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LETTER
Thermoacoustically driven flame motion and heat releasevariation in a swirl-stabilized gas turbine burner investigatedby LIF and chemiluminescence
W. Hubschmid Æ R. Bombach Æ A. Inauen Æ F. Guthe ÆS. Schenker Æ N. Tylli Æ W. Kreutner
Received: 1 July 2005 / Revised: 27 February 2008 / Accepted: 2 March 2008 / Published online: 26 April 2008
� Springer-Verlag 2008
Abstract Laser-induced fluorescence and chemilumi-
nescence, both phase-locked to the dominant acoustic
oscillation, are used to investigate phenomena related to
thermoacoustic instability in a swirl-stabilized industrial
scale gas turbine burner. The observed sinusoidal phase-
averaged flame motion in axial (main flow) direction is
analyzed using different schemes for defining the flame
position. Qualitative agreement between experimental data
and theoretical analysis of the observed flame motion is
obtained, interpreted as originating primarily from varia-
tion of the burning velocity. The heat release variation
during an acoustic cycle is determined from the sinusoi-
dally varying total OH* chemiluminescence intensity.
List of symbols
A0, A1 parameters of gas flow (Eq. A4)
BZ bottom flame zone
c adiabatic sound velocity
cfuel fuel concentration
CZ central flame zone
CL chemiluminescence
CM center of mass
D diameter of burner
d diameter of combustion chamber
Dhfm molar formation enthalpy
ICL CL intensity
LIF laser-induced fluorescence
pac acoustic pressure
D _Q heat release rate into a volume element
sT turbulent flame speed (flame velocity relative to
gas flow)
Sr ¼ mDu Strouhal number
SZ shear layer zone
Tair inlet air temperature
t time
TZ top flame zone
u inlet gas flow velocity
u0 turbulence intensity
DV volume element
vb burning velocity
vgas,0 stationary part of phase-averaged gas flow
velocity
vtotgas phase-averaged gas flow velocity
vr axial projection of turbulent flame speed
d�vr amplitude of variation of vr(t)
�vr temporal average of vr
vvar phase-dependent gas flow velocity
x, y transverse coordinates
DxDy cross section element of combustion chamber
z axial coordinate
zfl flame position in axial direction
z0 temporal average of zfl zfl,max (min) maximum
(minimum) of zfl Dzfl = zfl,max - zfl,min
_zfl tð Þ axial component of flame velocity in laboratory
a phase angle of maximal flame velocity relative
to gas flow
c ratio of heat capacities (cp/cv)
v phase angle of maximum of heat release rate
d phase shift between the variation of the heat
release and the OH LIF intensity f = A1/x0 angle between flame front and x–y plane
W. Hubschmid (&) � R. Bombach � A. Inauen � S. Schenker �N. Tylli � W. Kreutner
Paul Scherrer Institut, 5232 Villigen PSI, Switzerland
e-mail: [email protected]
F. Guthe
ALSTOM (Schweiz) AG, 5405 Baden-Dattwil, Switzerland
123
Exp Fluids (2008) 45:167–182
DOI 10.1007/s00348-008-0497-1
q0 average gas density
/ fuel–air equivalence ratio
u phase angle of forward zero crossing of flame
motion
w phase angle of maximum LIF intensity
x angular frequency of sound
1 Introduction
Modern industrial gas turbines are operated at very lean
premixed flame conditions in order to reduce the emis-
sion of NOx. In this range of operation the occurrence of
thermoacoustic instabilities is favored. The temporal
variation of the heat release rate is known to be the main
source of pressure waves in the system. Resonance of
the combustor and feedback of the pressure variation on
both the fuel/air mixing and the turbulence fluctuation
lead to selective amplification of distinct oscillation fre-
quencies. One important engineering goal is to devise a
theoretical description of the observed phenomena like
the periodic variation of the heat release and of the
burning velocity, and to allow predictions of the growth
conditions for pulsation modes and of the asymptotic
pressure level of the sound oscillation. Current theoreti-
cal descriptions use simplified models as a basis for
numerical calculations. Therefore, experimental informa-
tion on thermo-acoustically unstable combustion
processes is needed in order to validate the numerical
simulations, as well as to guide the further development
of burners and combustors and of the numerical codes.
Because of their non-intrusiveness, optical measurement
techniques, like laser-induced fluorescence (LIF),
chemiluminescence (CL), or particle imaging velocimetry
(PIV) have been widely employed to investigate com-
bustion instabilities in gas turbines.
A comprehensive overview on theory and phenomena of
instabilities in gas turbine engines is given in the recent
book edited by Lieuwen and Yang (2005). See here espe-
cially the articles by Ducruix et al. (2005), Huang et al.
(2005), Menon (2005), Lieuwen (2005), and Dowling and
Stow (2005). Earlier, Paschereit et al. (2000, 2001, 2002),
Kruger et al. (2001), and Krebs et al. (2002b), among
others, discussed specific problems of combustion insta-
bilities in gas turbines. A review on combustion dynamics
and its control can, e.g., be found in Candel (2002).
Experimental results on combustion instabilities, focussing
mainly on observations of the variation of the heat release
and the fuel concentration with the acoustic oscillation,
were reviewed by Lee and Santavicca (2005).
In order to have guidelines for experiments on ther-
moacoustic instabilities in combustion and to understand
and interpret the phenomena observed in the experiments, a
look at the theory of sound generation in combustion is
needed. The fundamental equation is the equation for the
generation and the propagation of sound waves. In the case
that only the variation of the heat release rate is considered
as source of sound, the wave equation has the following
form (Ingard 1968):
o2pac
ot2�r c2rpac
� �¼ c� 1ð Þ o
ot
D _Q
DVð1Þ
In Eq. (1) pac is the sound pressure, D _Q is the heat release
rate in the volume element DV = DxDyDz; it equals the
rate of change of sensible enthalpy, which is generated by
the chemical reaction, into this volume element. By con-
vention, x and y are the transverse coordinates, and z is the
axial coordinate, which coincides with the main flow
direction (see Fig. 2). Furthermore, the ratio c of specific
heats and the adiabatic sound velocity c appear in Eq. (1).
Other source terms, as the turbulence source term, are
neglected in Eq. (1), as they are assumed to be distinctly
smaller than the heat release term. The dissipation terms
are not written explicitly either. In the case of sound
pressure amplitudes that are not negligible compared to the
average pressure, a generalized non-linear sound equation
has to be applied instead of Eq. (1) (Poinsot and Veynante
2001).
The heat release rate per cross section element DxDy of
the combustor chamber is
D _Q
DxDy¼ �cfuel t; x; yð Þ vb t; x; yð Þ
cos JDhm
f : ð2Þ
In Eq. (2), vb(t, x, y) is the burning velocity, i.e. the volume
of reactant gas burnt per cross section unit of the flame
brush and per time unit (the flame brush, or shortly flame,
is assumed to be infinitely small), 0 is the angle of the
flame to the x–y plane, cfuel(t, x, y) is the fuel concentration
at the coordinates x and y of the flame, and Dhfm is the
(negative) molar reaction enthalpy at standard temperature.
According to Eq. (2), variation of local heat release may
originate from either variation of vb(t, x, y), cfuel(t, x, y),
and/or cos0, i.e. the (infinitesimal) flame area. Variation of
vb(t, x, y) with the sound pressure can originate from var-
ious sources. According to Filatyev et al. (2005), the
burning velocity vb(t, x, y) is closely related to the turbulent
flame speed sT, which is the speed of the flame front,
respectively, of some iso-line of the reaction variable,
relative to the flow. sT itself depends on the laminar flame
speed sL and on the turbulence intensity u0, which both vary
with the presence of sound waves: ‘The laminar flame
speed sL depends on the fuel concentration and on the
temperature, and therefore on the sound pressure; on the
other hand Poinsot et al. (1987) and Poinsot and Veynante
(2001) showed that sound waves may generate vortices and
168 Exp Fluids (2008) 45:167–182
123
thus enlarge the turbulence intensity u0. Explicit relations
between sT, sL and u0 are given by Zimont and Lipatnikov
(1995), Polifke et al. (2000), or Filatyev et al. (2005).
Variation of the fuel concentration with the fluctuating
pressure of acoustic waves originates from the effect that
fuel concentration is modulated by the variation of the
sound pressure at the injector of the burner (see Schuer-
mans et al. 2004). With some time delay the induced
variation of the fuel concentration reaches the flame.
Biagioli et al. (2007) indicate that from the unsteady flame
anchoring, sound pressure oscillations also may generate
variations of the infinitesimal flame area, i.e. of the angle 0.
The temporal variation of cfuel(t, x, y) can be observed
by either an absorption measurement or by observing the
LIF of a tracer added to the fuel. The axial projection vr of
sT(t, x, y, z) can be derived from LIF measurements of
flame species, as motion in axial direction of the flame. As
sT(t, x, y, z) is defined as velocity relative to the gas, for the
determination of sT(t, x, y, z) also the gas flow velocity has
to be measured. By observing the phase-averaged flame
zone, the angle 0 can be determined, too.
Lee et al. (2000) were among the first to determine the
temporal variation of the heat release in unstable combus-
tion. They determined the heat release in a lean premixed
model combustor by means of CL of OH*, and compared it
with values derived from the flame surface density as
obtained from OH LIF. The authors show that there is good
agreement between these distinct measurements. Ferguson
et al. (2001) applied high-speed OH* and CH* CL imaging to
stable and unstable conical flames in a Rijke Tube combustor
in order to determine the global heat release rate and the
flame surface area, respectively. Krebs et al. (2002a, b) and
Khanna et al. (2002) measured the flame response of swirl-
stabilized flames to externally driven oscillations. Using
OH* CL as a measure for the heat release rate they compared
flame response and incoming velocity fluctuations. Giezen-
danner et al. (2003) reported on the combination of quasi-
simultaneous phase-resolved LIF and CL measurements of
OH, CH, CH2O, and of OH* in order to observe phenomena
related to combustion instabilities in a swirl-stabilized dif-
fusion burner. Bellows et al. (2006) investigated the
saturation of the pressure fluctuations with growing heat
release by means of OH* and CH* CL.
Measurements of fuel concentrations in unstable com-
bustion have been performed by Venkataraman et al.
(1999), who combined phase-resolved CH* CL with fuel/
air mixing data obtained by means of LIF measurements of
the fuel tracer acetone. They conclude that both, changes in
flame area and fluctuations of equivalence ratio drive the
combustion instability in their lean premixed bluff-body-
stabilized combustor. The technique of acetone LIF has
been applied also by Schneiders et al. (2001) to investigate
the mixing processes in gas turbine burners.
The focus of this article is the investigation of the phase-
averaged periodic flame motion (phase-averaged: measure-
ments averaged at fixed phase of the investigated acoustic
oscillation), which is the consequence of the periodic vari-
ation of the burning velocity in thermo-acoustically unstable
combustion (cp. comment to Eq. 2). Such flame motion was
observed in an industrial scale swirl-stabilized gas turbine
burner. Phase-locked LIF of flame species was employed to
localize the flame zone and to detect its motion at the pre-
valent frequency of the acoustic oscillation (Sects. 3.2, 3.3).
External sound forcing ranging at levels from zero to max-
imum power of the loudspeaker system was applied in order
to stabilize the acoustic oscillation rendering phase-locking
easier. To determine the position of the flame zone, a
simultaneous measurement of OH and CH2O would be
appropriate (Paul and Najm 1998). For the observation of the
flame motion, however, the absolute value of the flame
position is not of first relevance; thus each of the two species
alone can well be employed. The radical CH has a rather
small concentration at the lean conditions used in gas turbine
combustion, so that its use in LIF measurements is excluded.
In our investigations, most of the measurements used OH as
indicator for the flame zone; in a few cases, however, the
CH2O distribution was determined to obtain the flame
motion. LIF of CH2O is advantageous compared to LIF of
OH, if, in addition to the flame position, the phase angle of
the maximum of the heat release rate should also be deter-
mined. It is probable that the determination of this phase
angle by means of LIF of CH2O results in a more accurate
value for this quantity (cp. the discussion in Sect. 3.2). The
phenomenon of phase-averaged flame motion was previ-
ously reported at various conferences, cp. Hubschmid et al.
(2002) and Schenker et al. (2003). Hassa et al. (2002)
observed flame motion by means of OH* CL in a liquid fuel
combustor operated at pressures up to 20 bar. In a premixed
swirl-stabilized combustor a periodic motion of the flame
was observed by Bellows et al. (2007). Different techniques
for quantifying the phase-averaged motion of the flame are
presented and compared in this paper. Also, the phase rela-
tion between the flame motion and the periodic variation of
the heat release rate was determined for a large number of
operation conditions. In additional measurements, CL of
OH* was applied for a quantitative determination of the
variation of the heat release (Sect. 3.4). Furthermore, by
means of LIF of acetone the modulation of the fuel con-
centration by the acoustic oscillation was observed
(Sect. 3.5).
2 Experimental set-up
The experiments with OH LIF and OH* CL were per-
formed on a commercial single swirl-stabilized gas turbine
Exp Fluids (2008) 45:167–182 169
123
burner (Alstom EV burner), which was operated at atmo-
spheric pressure with a power of about 700 kW. A
schematic drawing of the EV burner is shown in Fig. 1.
The burner has been described in detail by Dobbeling et al.
(2005). Figure 2 presents a schematic of the burner and
combustor together with the optical experimental set-up.
The fuel/air supply, upstream of the burner and the exhaust
are not shown. Natural gas is injected both along the air
slots of the burner cone and from a lance situated in the
cone axis. The air is preheated (673 K \ Tair \ 743 K) and
enters from the plenum through the slots, which separate
the two half-cones of the burner. The temperature of the air
was measured by thermocouples right after the electric pre-
heaters and prior to the burner and fuel–air mixing section.
Fuel and air mix in the burner cone and enter the cylin-
drical combustion chamber with a high swirl. After the
sudden expansion into the combustion chamber (about 1:4
in area), recirculation zones are formed in the central and in
the outer regions. These are confirmed by numerical sim-
ulations (Flohr et al. 2002; Biagioli 2006), as well as by
LDA measurements in water flow experiments (Paschereit
et al. 1999) and by PIV measurements (Schuermans et al.
2006). A section of the combustion chamber, from the
burner exit plane to about one chamber diameter down-
stream, is made of UV transparent quartz glass and cooled
by air.
Depending on the operating conditions of the burner,
acoustic oscillations at distinct frequencies are established
inside the combustion chamber, one of them at about
Sr = 0.51 being clearly dominant. The sound pressure in
the combustion chamber was measured with a calibrated
water-cooled microphone positioned close to the flame
zone, i.e. approximately 1.5 chamber diameters down-
stream of the burner exit plane. Forcing the dominant
frequency with two sets of four loudspeakers each (total
power of 1,200 W) enhanced the continuity of the com-
bustion instability. This facilitated triggering of the laser
pulses and was applied for most of the experiments. One of
the loudspeaker sets was mounted upstream of the burner in
the air plenum and the other one at the exhaust. An example
of an unforced experiment is described in Fig. 5a, b.
For the OH LIF experiments, the frequency-doubled
light of a pulsed Nd:YAG/dye laser system (Quantel
YG781/TDL50) at 286 nm was used to excite the P1(7) line
of the A / X (1 / 0) transition of the OH radicals. This
line is rather insensitive to temperature variations in the
range considered. The repetition rate of the Nd:YAG laser
was fixed to 20 ± 0.2 Hz. A specially designed phase-lock
loop (PLL) filter allowed for locking of the laser repetition
rate to the dominant thermo-acoustic frequency. By the
experimental conditions the value of the phase-angle of the
laser pulses with respect to the phase angle of the acoustic
oscillation is defined only up to an unknown constant. The
period of the acoustic oscillation was divided into 24
intervals, i.e. 15� phase resolutions. At each phase angle of
one measurement series, 100 single pulse images as well as
10 images of 100 pulses summed up on the CCD chip were
recorded. The laser beam was shaped by appropriateFig. 1 Schematic drawing of the Alstom EV burner
Fig. 2 Scheme of the burner, combustor, together with the optical
experimental set-up
170 Exp Fluids (2008) 45:167–182
123
combination of cylindrical and spherical lenses into a
slightly divergent light sheet with a width of about 110 mm
and a thickness of approximately 0.2 mm, propagating
from bottom to top through the combustion chamber tube.
The laser light sheet was oriented in the y–z plane, y being
the propagation direction of the laser light, and z being the
symmetry axis of the tube, i.e. the main flow direction (see
Fig. 2). The OH LIF signal was detected normal to the
plane of the light sheet using an intensified CCD camera
(LaVision FlameStar II, gate: 40 ns) equipped with a band
pass filter centered at 310 nm (FWHM 14 nm). The filter
chosen allowed recording several rotational lines of the
(1 ? 0) transition of OH. All images were corrected for
camera background.
For most of the investigated operation conditions, in
addition to OH LIF, also 2-D line-of-sight CL of OH* was
registered. Phase delay and camera set-up were the same for
the OH* CL measurements; the camera exposure time was
0.1 ms. The detection area in flow direction was larger for the
CL measurements as compared to the LIF recordings, as they
are not limited by the width of a laser light sheet. In the
spectral range of the band pass filter used, one has to expect
also CL of CO2* (Nori and Seitzman 2007). CO2* shows
broadband emission resulting in a minor addition to the
detected OH* CL only. This does not disturb the final results,
as we rely always on relative intensities.
The supplementary experiments with LIF of formalde-
hyde (CH2O) were performed with a modified EV burner,
and with a modified combustion chamber, which is closer
to the configurations in a real gas turbine. Especially, the
stainless steel combustion chamber was of rectangular
shape and was water-cooled. This design allowed for
mounting of one set of four loudspeakers (total power
600 W) in the air plenum upstream of the burner mixing
section only. Optical access was provided by two large
quartz windows, embedded into the bottom sidewall of the
combustion chamber. They are situated as close as possible
to the burner edge where sudden expansion takes place.
Several water-cooled microphones allowed for the mea-
surement of both, the frequency and the amplitude of the
acoustic oscillations. For CH2O LIF the frequency doubled
output of the same laser system, except for the dye (pyri-
dine 1), was used to excite into the A / X 401 transition at
around 352.5 nm. Fluorescence of several vibronic emis-
sion bands in the spectral range of 400–550 nm was
detected using the same camera type equipped with a
combination of Schott filters (KV389 and BG39/3).
The fuel–air mixing and the influence/effect of the
modulation of the sound wave were investigated by LIF
measurements of acetone, which was added to the fuel
prior to mixing as tracer. These experiments were also
carried out with a modified EV burner. This burner had an
additional mixing section, partly made of UV transparent
quartz, prior to the sudden expansion into the combustion
chamber. The laser light sheet entered from below covering
the entire length of the transparent section along the sym-
metry axis. Acetone was excited using the fourth harmonic
of the Nd:YAG laser at 266 nm. Typical tracer mole
fractions were below 2%. This low concentration ensured a
complete vaporization of the acetone prior to its entering
the burner section. Tracer concentration studies showed no
saturation effects. Therefore, incomplete vaporization
could be excluded. The fluorescence was collected normal
to the plane of the laser-light sheet with the camera
described above, and equipped with a long pass filter
(Schott WG295) to separate the LIF signal from reflected
and/or (Rayleigh) scattered light.
In identical operating conditions of the burner, sound
pressures in the combustor were clearly higher with
acoustic forcing than without forcing. Full forcing resulted
in up to four times higher sound pressures than without
acoustic forcing. Partial forcing, i.e. only upstream forcing
at 70% of full power of the loudspeakers, resulted in about
1.3–1.7 times higher sound pressures than without. For the
same power levels of the loudspeakers, the sound pressures
for flows with a flame were by far larger than the ones for
flows of preheated air only, where the fuel was replaced by
N2. For a typical operation condition, a peak sound pres-
sure of 800 ± 50 Pa was measured for the flame compared
to 40 Pa for the airflow only. Thus, the induced sound
intensity by the loudspeakers is by far smaller than the
sound intensity generated by the unstable burning flame.
The levels of naturally occurring sound pressure were
distinctly lower in the burner and combustor used for
the CH2O LIF experiments than in the original burner
and chamber design. Typical peak pressure oscillations
of \300 Pa were observed. In addition, the influence of
forcing was significantly lower due to the combustor design
as well as the lower forcing power applied. This compli-
cated the phase-locking of the laser diagnostics equipment
to the dominant acoustic oscillation. We suppose that this
is the main reason for the observed reduced sinusoidal
dependence of the LIF intensity on the acoustic phase. In
the meanwhile, an alternative to the phase-locking tech-
nique is proposed by Guthe and Schuermans (2007) for
circumventing this difficulty. They determine the acoustic
phase of each laser pulse and group measurements with
similar acoustic phases.
3 Results and discussion
3.1 Flame zone from LIF and from CL
For some given operation condition, Fig. 3a shows a
selection of single pulse LIF images of the OH distributions.
Exp Fluids (2008) 45:167–182 171
123
The naturally occurring thermoacoustic oscillations were
forced here externally. The images were all taken at one
arbitrarily chosen phase angle of the dominant acoustic
oscillation. Their vertical extensions correspond to about
the diameter of the combustion chamber, apart from a small
missing region at the bottom, which was masked by a bolt.
The signal distributions and shapes vary strongly from pulse
to pulse, and there is no characteristic pattern observed for
any of the phase angles considered.
Averaged images of the flame zone, for fixed phases of
the dominant acoustic oscillation, reveal a more regular
behavior. A periodic variation of the OH LIF intensity and
a spatial motion of the OH zone are observed in the case of
acoustic forcing as well as without any forcing. Figure 3b
shows the phase-averaged OH LIF image for 100 single
pulse images of the experiment of Fig. 3a. In Fig. 4a the
development of this flame in intervals of 45� of the phase
angle is shown. The section of the OH distribution
visualized by the laser light sheet is a v-shaped domain in
the center, with an opening angle of about 40� with respect
to the plane normal to the main flow direction. Additional
regions of OH are seen in the outer re circulation zones.
For the same operation conditions, also OH* CL was
recorded. Figure 4b, c show the results of the phase-
resolved (line-of-sight) CL measurements for the same
circumferential orientation of the burner as in the LIF
experiment (Fig. 4a) and with the burner rotated by 90�,
respectively. The axial asymmetry of the design of the
burner is recognized. A general asymmetry of the intensity
distribution, i.e. higher CL intensity in the upper part of the
figure, is distinctly observable.
For the analysis of the phase-averaged OH LIF mea-
surements, the 2-D distribution of the OH LIF intensity was
split into small sub-zones of about 1/30th of the chamber
diameter. Some sub-zones both in the center and in the
outer zones yielded similar temporal behavior of the LIF
Fig. 3 a Selection of four
single-pulse LIF intensity
distributions of OH at phase 0�of the dominant acoustic
oscillation. b (rightmost figure)
OH LIF image averaged at
phase 0� with classification of
the flame sections, i.e. top (TZ),
central (CZ) and bottom flame
zone (BZ), and the transient
flame zones in between. The gas
flow is from left to right.Forcing was applied to the
acoustic oscillation
a)
b)
c)
Fig. 4 a Phase-averaged OH LIF intensity measurements at the
phases of 0�, 45�, 90�, 135�, 180�, 225�, 270� and 315� of the
dominant forced acoustic oscillation. b Corresponding phase-averaged
OH* CL measurements for the same operating conditions. c Phase-
averaged OH* CL measurements with the burner rotated by 90�
compared to b. For both series of CL, the camera position was shifted
downstream by about d/8 compared to a. Therefore the images in
b and c are displayed with an offset with respect to those in a so that
corresponding points in the combustion chamber lie on the same
vertical line
172 Exp Fluids (2008) 45:167–182
123
intensity and the flame motion. Therefore, these sub-zones
were combined to larger sections: in a central flame zone
(CZ) and two outer zones close to the burner wall, named
bottom (BZ) and top flame zones (TZ; ‘‘bottom’’ and ‘‘top’’
refer to the set-up geometry; see Fig. 3b). BZ and TZ
coincide about with the outer recirculation zones of the
flow field. The temporal behavior in BZ and TZ differs
strongly from the one in CZ, with respect to phase angles
and amplitudes of the observed flame motion and variation
of the LIF intensity. The temporal behavior in the transient
zones in between marks the transition from central to
peripheral zones.
The laser light sheet positioned close to the burner exit
plane covered the main heat release zone and thus, the
main domain of OH formation. It was shown by mea-
surements shifted downstream by about the sheet width
that in smaller concentrations OH is found also more
downstream, especially in the outer recirculation zones. In
the central zone, however, the OH concentration is very
small.
From the comparison of the OH* CL with the OH LIF
results (see Fig. 4a), we conclude that attenuation of the
laser light along the propagation of the light sheet was on
the order of not more than 10%. Actually, for some addi-
tionally measured operation conditions, intensity of OH
LIF was even somewhat larger in the TZ than in the BZ.
This is in accordance with the OH* CL results: the integral
OH* CL intensity in TZ was larger by about 10% than in
the BZ. Thus, for the quantities of interest, as phase-
averaged variation of OH LIF intensity, laser light atten-
uation can be neglected.
In Fig. 5a, c, the difference between operation without
and with sound forcing is demonstrated. All other operation
conditions were the same, namely Tin = 723 K, / = 0.48,
and u = 40 m s-1. In the case of acoustic forcing, the
observed peak pressure amplitude was larger by a factor of
1.9: 930 versus 500 Pa. Figure 5a, c show the phase-
averaged OH LIF intensities as a function of the oscillation
phase and integrated over the central flame zone CZ. We
observe a variation in OH LIF intensity of about ±6% and
±17%, respectively, during one oscillation period, with a
slight change of the phase for the maximum of intensity.
For each of the three zones CZ, BZ and TZ, a sine function
can well be fitted to the phase averaged and spatially
integrated OH LIF intensity data. In the transition zones the
intensity variation is reduced and the phase is shifted
somewhat, compared to the CZ, whereas in the BZ and TZ
averaged LIF intensities vary stronger again. The standard
deviations of single-pulse intensity fluctuations, divided by
the mean intensity, at a given phase and integrated for CZ,
were in the forced case between 26 and 32% for the various
phase angles. Therefore, standard deviations of the mean
values are about 3% for N = 100 pulses. This agrees
approximately with the deviations of the mean values of
the various phases from the sinusoidal fit curve of Fig. 5c.
3.2 Flame motion from phase-averaged OH and CH2O
LIF measurements
As the flames in gas turbine burners are rather spread out,
various quantities can be defined to represent their position
and motion. For the observation of the flame motion, the
absolute value of the flame position is of minor importance;
-4
-2
0
2
4
cent
er o
f mas
s [m
m]
36031527022518013590450
phase angle [deg]
1.2
1.0
0.8
LIF
inte
nsity
[a.
u.]
-4
-2
0
2
4
cent
er o
f mas
s [m
m]
1.2
1.0
0.8
LIF
inte
nsity
[a.
u.]
a)
b)
c)
d)
Fig. 5 Variation of a the OH LIF intensity and b the axial coordinate
of the center of mass for the OH LIF intensity in the central zone
during an oscillation period and their corresponding sinusoidal fit
(solid line) for an experiment without sound forcing. c, d Respective
quantities for experiment with same operation conditions as
mentioned above, but with sound forcing
Exp Fluids (2008) 45:167–182 173
123
rather a relative value that includes an arbitrary offset is
required. In a first definition, the flame position is identified
with the phase-averaged center of the OH LIF intensity in
flow direction, here called center of mass (CM). This
quantity is averaged along the radial coordinate y, i.e.
along the light sheet propagation, for the CZ, as well as for
the BZ and TZ of the combustion chamber. For the
experiments described in the last paragraph of the previous
sub-section, Fig. 5b, d show the motion of CM in the CZ as
a function of the oscillation phase, without and with sound
forcing, respectively. A sine function can well be fitted to
the phase-averaged data, spatially integrated for each of the
zones CZ, BZ and the TZ. The total variations of the flame
position (CM) are Dzfl = 3.0 and 6.4 mm, respectively.
For the (forced) experiment of Fig. 4a, results for the
flame motion in the sub-zones of CZ are presented in
Fig. 6. They show that the phase-averaged flame motion
has the character of a piston motion. One recognizes from
the figure that the angle 0 between flame front and x–y
plane does not vary strongly with the acoustic oscillation.
A few repeated series of phase-averaged measurements
under identical experimental conditions were performed.
Deviations of not more than 15� for the phase angles for
minimum flame position and maximum OH LIF intensity
were found, which gives an indication on the accuracy of
the determination of phase angles.
The phase-relation between the variation of the OH LIF
intensity and the flame motion was studied by variations of
inlet temperature, inlet flow velocity, global fuel–air ratio
and mode of fuel injection. A correlation, however weak,
between the phase angles of the OH LIF intensity and of the
flame motion was found. For 17 out of the 19 investigated
cases the maximum OH LIF intensities in the CZ occurred at
phase angles that were between 75� and 132� larger than the
ones of the backward zero crossing of the flame motion.
Thus, they turned out to be at about the phases of minimum
flame position. (The other results differed by ±10� more.) In
order to check the relation between the phase angles of the
variation of the OH LIF intensity and the flame motion, an
additional (acoustically forced) experiment at a slightly
different burner (cp. Sect. 2) was performed, in order to
determine the distribution of CH2O by LIF. Figure 7a shows
a selection of single pulse LIF images of CH2O; an image of a
CH2O measurement averaged at a given phase is shown in
Fig. 7b. Figure 7c shows the variation of the CH2O intensity
during the acoustic period in the zone noted in Fig. 7b
(‘‘shear layer zone’’, SZ). From the experiment with CH2O
LIF, we found, in the SZ, a shift of 70� between the phase
angle of backward zero crossing and the phase angle of the
maximum of the CH2O LIF intensity (Fig. 7c).
The findings that the phase angles of maximum LIF
intensity and minimum flame position correlate in a given
flame zone, reflects the relation between heat release rate
D _Q; burning velocity vb, and fuel concentration cfuel, see
Eq. (2). In the data analysis performed here, LIF intensities
are considered to provide an approximate measure for the
heat release rate. Careful interpretation is needed specifi-
cally when considering LIF intensities of OH as a measure
for the heat release rate. The example of calculations for
atmospheric laminar premixed combustion show (Glass-
mann 1996) that the OH concentration rises approximately
proportional to the temperature at the boundary between
fresh and burnt gases, but decreases slowly towards equi-
librium afterwards, whereas the temperature still increases
slightly. The local heat release rate can therefore not be
obtained from the local OH LIF intensity.
In lean hydrocarbon flames OH is produced during the
combustion process. The OH concentration initially
surmounts the thermal equilibrium concentration, which is,
however, quickly reached after the flame front. To a first
-0.15 -0.15
-0.10 -0.10
-0.05 -0.05
0.00 0.00
0.05 0.05
0.10 0.10
0.15 0.15
radi
al p
ositi
on [
1/d]
axial position [1/d] ∆zfl
/d
0 45 90 135 180 225 270 315 360
a) b)
0.1920.192
0.1920.192
0.192 0.192 0.1920.192 0.01 0.020.192
Fig. 6 a Radial positions of the
center of mass of the OH LIF
intensity in the central zone
versus the axial position for one
period of the dominant acoustic
oscillation. The curves for
subsequent phases are plotted
with an increasing horizontal
offset with respect to the curve
at the phase of 0�. b Total
motion of the flame in the
central zone given as the
amplitude of the center of mass
between minimum and
maximum axial positions
174 Exp Fluids (2008) 45:167–182
123
approximation the thermal equilibrium concentration of
OH depends linearly on the concentration of H2O in the
exhaust gas region, and exponentially on temperature. The
concentration of H2O depends on its part linearly on fuel-
to-air ratio, whereas the temperature is a nonlinear function
of /. Assuming complete combustion in lean flames, the
heat release rate is directly related to the fuel-to-air ratio.
The observed OH concentration is therefore related to the
heat release, albeit in a complex form.
If one assumes in a model calculation that the OH
concentration decays exponentially after its formation, a
phase shift d between the variation of the heat release and
the OH concentration results. It lies between 0� and 90�,
depending on the ratio of the constant of decay to the
angular sound frequency x. The phase shift d is reduced if
the OH is transported out of the observation volume, as it is
the case for OH in the central zone. The phase shift
was observed by Giezendanner et al. (2003) in a swirl-
Fig. 7 a Selected single pulse
CH2O LIF images at one phase
angle (above, left). b CH2O LIF
image averaged over all phases
(above, right). c Variation of
both the CH2O LIF intensity
and center of mass and their
corresponding sinusoidal fits
(solid line) during one period of
the dominant oscillation for the
zone indicated with dashed lines
in b (shear layer zone, SZ)
(below). Acoustic forcing was
applied in this experiment
Exp Fluids (2008) 45:167–182 175
123
stabilized diffusion flame burner, comparing the phases of
OH LIF measurements with those of CH2O LIF, CH LIF,
or OH* CL.
The main step in the analysis is to relate the flame
motion in the inhomogeneous flow field, as observed in the
laboratory, to the variation of the flame velocity relative to
the gas flow. This is done in the ‘‘Appendix’’, using a
number of assumptions and approximations, namely:
neglect of phase-dependent gas flow velocity; homogeneity
of the flow field in the combustor in the transverse axes (in
x–y plane); time-averaged flow direction parallel to z-axis;
stationary external conditions; infinitely small thickness of
the flame brush in propagation direction, and therefore
equality between turbulent flame speed and burning
velocity. The calculation shows that in the case of a spa-
tially inhomogeneous flow with a large negative gradient of
the mean flow velocity, the phase of the maximum flame
velocity relative to the gas is at about the minimum flame
position. On the other hand, in the case of a homogeneous
flow field, the maximum flame velocity relative to the gas
is at the backward zero crossing of the flame motion. The
assumption that the gas flow velocity does not depend on
the acoustic phase is an approximation. A phase-dependent
modulation of the gas flow velocity is expected from the
acoustic velocity. We refer here to Duan et al. (2005), who
determined a periodic oscillation of the gas flow velocity in
a gas turbine model combustor exhibiting thermoacoustic
instability by means of phase-resolved laser Doppler
anemometry.
Under the additional assumption that the phase differ-
ence between oscillations of cfuel and vb is small, we
conclude that the phase angle v of maximum heat release
rate has to be in the interval between backward zero
crossing and minimum flame position. The assumption is
justified if the main contribution to the variation of vb is
caused by variation of cfuel. In the case of OH LIF the
phase angle w of the maximum LIF intensity is enlarged by
the angle d, compared to v. Considering all the above-
mentioned effects, we find qualitative agreement between
theoretical analysis and experimental data on the phase
relation between the oscillations of the LIF intensity and of
the flame motion. To get the order of magnitude for vari-
ations of the burning velocity, one may make the
approximation of a homogeneous velocity field and neglect
the phase-dependent gas flow velocity. For the flame
motion of Fig. 5d, the calculation in the ‘‘Appendix’’ gives
a variation of the flame velocity in axial direction d�vr ¼0:06 u (u is the inlet gas flow velocity), if one inserts the
total amplitude of the flame motion as obtained from the
CM definition of flame position.
Instead of the center of mass, the rising slope of the OH
LIF intensity profile could also be used as a measure for the
flame position. Figure 8 shows the normalized profiles for
the central zone of the phase-averaged OH LIF intensity
along the axial burner direction at the phase angles of 135�and 315� for the (acoustically forced) OH LIF series of
Fig. 4a. The two phases represent about minimum and
maximum positions of the flame. Plotting the axial posi-
tions of the rising edge versus the oscillation phase for the
values 0.3–0.8 (in intervals of 0.1) of the OH LIF inten-
sities, normalized to their maximum, one finds
Dzfl = 9 mm as a mean value, whereas the corresponding
value for the motion of CM was Dzfl = 5 mm. Figure 8
indicates that in the downstream part the difference in
position of the OH regions diminishes for the different
phase angles. Therefore, a difference in the values for Dzfl,
using different definitions of the flame position, is expec-
ted. However, the phase of minimum flame position, i.e.
the position closest to the burner edge, turned out to be
unambiguous. Within the accuracy of the determination,
the same values for both definitions of the flame position
were obtained.
3.3 Flame motion derived from single-pulse
OH LIF images
Instead of deriving the flame motion from phase-averaged
measurements one may proceed by determining a flame
front trace for each single-pulse image. This evaluation is
based on the fact that the OH concentration rises steeply in
the main heat release zone. In the present work, a median
filter was applied iteratively to the raw images (of the
experiment of Fig. 4a) in order to smooth out smaller
holes. Each smoothed image was thereafter binarized with
a threshold set at the level of 15% of the maximum image
intensity value. The outline of the flame was obtained in
1.0
0.8
0.6
0.4
0.2
0.0
norm
aliz
ed L
IF in
tens
ity
0.350.300.250.200.150.100.050.00axial coordinate [1/d]
phase angle: 135 315
Fig. 8 Profiles of OH LIF intensity along the axial direction for the
central zone at the phases 135� and 315� of the dominant acoustic
oscillation
176 Exp Fluids (2008) 45:167–182
123
the binarized image by applying a morphological opera-
tion, which returns 0 when all four connected pixels have
the same value (either 0 or 1). Figure 9a shows a repre-
sentative single-pulse OH LIF image; the diameter at the
exit of the swirl burner is about 55% of the image height;
Fig. 9b shows the corresponding flame front trace, obtained
by applying the binary image processing scheme.
Performing these operations for all 100 single-pulse
images of a given phase of the dominant thermoacoustic
oscillation, and adding the resulting flame fronts, an image
of the distribution of the flame fronts is obtained.
Figure 10a shows the images of the flame front distribu-
tions for a resolution of the phase angles of 45�. Average
flame front motion can clearly be recognized. It should be
noted that both the rise and drop in image intensity give a
contribution to the flame front image. However, as these
regions are typically well separated, this is not a serious
limitation, and this analysis of the data is free from an
artificial cutting off of lines. In order to determine quan-
titatively the motion of the flame, the central section of the
image was subdivided into 10 pixel high horizontal slices.
For each of these slices, the contributions of flame front
traces in the vertical extension (i.e. over the 10 pixels)
were added to give an intensity function along the image
width. The first hill after the burner exit was subsequently
fitted with a Gaussian, and for each slice the axial position
corresponding to the maximum value of the Gaussian was
chosen as the flame position. In order to reduce the noise,
the flame front positions of four such slices were averaged.
In this way the flame position in the central part of the
flame was determined. Figure 10b shows the variation of
the flame position over a period of the dominant ther-
moacoustic oscillation. The phase angle of the minimum
flame position in the central part of the flame is very close
to the one derived from the motion of the center of mass in
CZ. The amplitude of the sinusoidal motion obtained is
Dzfl = 13 mm, and thus, larger than the value from the CM
motion in CZ and also larger somewhat than the value
obtained from the rising slope in the phase-averaged ima-
ges in CZ (cp. Sect. 3.2, last paragraph).
3.4 Global heat release variation from OH* CL
Integration of the phase-averaged 2-D OH* CL signals, see
Fig. 4b, c, over the whole flame yields a measure (see
below) for the total heat release rate at a specific phase
angle. Consistent with the LIF measurements, the total CL
intensity also varies sinusoidally during the period of a
Fig. 9 a Representative single
pulse OH LIF image (left).b Flame front trace resulting
from a (right) )
Exp Fluids (2008) 45:167–182 177
123
sound oscillation. The observed CL intensity variations
over one oscillation period are ±14 and ±13%, respec-
tively, for the two orientations of the burner, respectively,
differing in azimuth angle by 90�. The phase angles of the
maxima of the CL intensity differed by 21�. In fact, the
measurements at the two burner orientations determine
both the total CL intensity, i.e. the same quantity. The
difference in the observed phase angles lies within the
accuracy of determining phase angles (i.e. about ±10�, cp.
Sect. 3.2). The result of this measurement, as well as of 10
other measurements with OH* CL (for totally 6 different
operation conditions), and of the measurement with CH2O
LIF (cp. Sect. 3.2) give intensity maxima at phase angles
between 5� and 36�, with one exception only (55�). Thus,
a)
b)
Fig. 10 a Superposition of
phase-resolved flame front
traces for phases of 0�, 45�, 90�,
135�, 180�, 225�, 270� and
315�. b Flame motion as
function of phase angle obtained
by technique of flame front
tracing
178 Exp Fluids (2008) 45:167–182
123
changing the mode of injection, or other operation condi-
tions, moderately changes the phase relation to the acoustic
oscillation. The phase relation indicates whether the heat
release amplifies or weakens the acoustic oscillation
(Rayleigh criterion).
We assume that a relevant if not overwhelming part of
the heat release variations in thermoacoustic instabilities
results from variations of the fuel–air equivalence ratio /(cp. Sect. 3.5). In order to infer the heat release variation
from the observed variations of the OH* CL intensities, we
performed a series of OH* CL measurements with varia-
tion of the global /, and averaging over measurements at
arbitrary acoustic phase angle. To cover the domain of
interest for the thermoacoustic instabilities in the investi-
gated burner, intensities ICL(/) for the range of / between
0.42 and 0.51 were determined, see Fig. 11. Approxi-
mately, a linear relationship between / and ICL(/) was
observed. This domain of / was not measured in the
previous investigations of Higgins et al. (2001), Lee and
Santavicca (2003), and Nori and Seitzman (2007).
The relation for the phase-resolved experiments between
the variation of the normalized OH* CL intensity and the
fuel–air equivalence ratio, and accordingly the variation of
the heat release rate on the normalized OH* CL intensity,
can now be determined approximately from the inverse
derivative b of the normalized function ICL(/) at /0, which
is the average value of / during the variation:
/� /0
/0
¼ bICL /ð Þ � ICL /0ð Þ
ICL /0ð Þ: ð3Þ
ICL(/0) and ICL(/) are the intensities of OH* CL for /0
and for a value / close to /0, respectively. For the mea-
surements given in Fig. 11, values for b between 0.25 and
0.27 result, depending on /0.
Taking into account these values of b, a variation of
±13.5% in the OH* CL intensity corresponds to a variation
of about ±3.5% in the heat release rate during an oscilla-
tion period. For other operating conditions, variations of up
to ±35% of the OH* CL intensity were observed, that is,
variations of heat release rate of up to about ±9%.
According to the plane wave solution of Eq. (1), sound
waves with peak pressure amplitudes of up to 2 9 102 Pa
are generated in the flame zone by these variations of the
heat release rate. These sound waves are superposed to the
sound waves formed previously, and in a quasi-steady state
they compensate for the dissipative losses of sound and for
the radiation of sound out of the combustion chamber.
3.5 Modulation of fuel concentration
To check whether there is a feedback from the sound wave
to the fuel injection, as assumed in the previous sub-sec-
tions, a (acoustically forced) tracer LIF experiment with
acetone, added to the fuel, was performed. For this
experiment a slightly modified burner configuration with
optical access to the fuel/air mixing section was employed.
Figure 12 shows the phase-averaged LIF intensity as a
function of the phase angle. During one period of the sound
oscillation the LIF intensity varies by about ±6%, which
proves that the fuel injection is modulated by the sound
wave for the given type of gas turbine burner. The observed
temporal variation of cfuel in the mixing region propagates
to the flame zone where it causes a periodic variation of the
heat release rate. This coupling of the sound wave back to
the flame is one of the important feedback mechanisms to
build up thermoacoustic instability in the given burner
type.
2.0
1.5
1.0
I CL/
I CL,
φ = 0
.42
0.520.500.480.460.440.42fuel-air equivalence ratio φ
Fig. 11 Normalized OH* chemiluminescence intensity ICL experi-
mentally determined as a function of the overall fuel–air equivalence
ratio /
Fig. 12 Phase-averaged tracer LIF measurements of acetone—added
to the fuel—in order to study the fuel modulation during an oscillation
period and its sinusoidal fit (solid line)
Exp Fluids (2008) 45:167–182 179
123
4 Summary and conclusions
We showed that phase-locked 2-D LIF and CL techniques
are useful tools for the investigation of thermo-acoustic
phenomena in commercial gas turbine burners, and thus for
the further development of gas turbine burners. Main
effects of acoustic instability studied in this paper are the
phase-averaged flame motion and the phase-averaged var-
iation of the heat release. Phase-locking was applied at the
dominant frequency of acoustic oscillation. Phase-resolved
data of the flame motion were obtained from phase-aver-
aged images as also directly from single-pulse images. In
order to distinguish the specific behavior of various zones
of the flame, quantities were in part determined separately
for the outer recirculation zones and the central flame zone.
Flame motion, together with data on the time-dependent
gas flow velocity, gives information on the variation of the
burning velocity in unstable combustion.
We observed for a large number of operating conditions
that the phase-averaged OH LIF intensity and the flame
position of the various flame zones vary sinusoidally with
the acoustic frequency. Oscillations are observed in the
case of external sound forcing as well as without sound
forcing. For identical conditions, phase angles of flame
motion and OH LIF intensity yielded reproducible values
within ±10�. Measurements with phase-averaged CH2O
LIF also showed a sinusoidal behavior of intensity varia-
tion and flame motion.
The amplitudes of flame motion in axial (i.e. the main
flow) direction depend strongly on the definition of the
flame position, i.e. center of mass versus rising slope in
phase-averaged images, or flame front trace from single-
pulse images. Phase angles of the flame motion, however,
do not depend on the accuracy of measurement on the
choice of definition. For the flame motion, interpreted as
originating mainly from the variation of the burning
velocity, we obtained qualitative agreement between
experiment and theoretical analysis.
Phase-averaged OH* CL intensities integrated over the
whole flame region vary sinusoidally by about ±13 to
±35% around their mean values, depending on the specific
operating conditions. By variations of the global fuel–air
equivalence ratio / we showed that there is an approxi-
mately linear dependence between variations of OH* CL
intensities and variations of heat release in the relevant
domain of very lean combustion with natural gas. From
these calibration measurements it can be concluded that in
the observed unstable combustion, phase-averaged heat
release rates, spatially integrated, varied in the range of
±3.5 to ±9%.
One important feedback mechanism in the formation of
the thermoacoustic instability in the given burner type is
the modulation of the fuel concentration cfuel by the
acoustic oscillation. This was shown by phase-locked LIF
measurements of acetone in the mixing part of a slightly
modified burner. The acetone had been added to the fuel as
a tracer.
Acknowledgments Financial support by the Swiss Federal Office of
Energy (BFE) and KTI (Kommission fur Technologie und Innova-
tion) is gratefully acknowledged. We thank O. Paschereit for planning
and supervising the experiments, M. Zajadatz, R. Lachner, C. Motz,
B. Paikert, T. Guntern for help during the experiments, especially for
running the Alstom test rig, and P. Flohr, B. Schuermans, F. Biagioli,
all of Alstom (Schweiz), V. Bellucci of PSI, and M. Furi of ETH
Zurich for helpful discussions.
Appendix: Quantitative approach to the phase-
averaged flame motion in a simplified model
We consider the heat release zone to be a volume char-
acterized in transversal extent by the coordinates x and y. In
case of a piston-like motion of the flame, as we assume it,
and which is suggested by experiment, the flame moves
back and forth along the combustion chamber axis. To
simplify the analysis, the (time-averaged) flow direction is
supposed to be parallel to this axis. Actually, this is not the
case for the flow field of the investigated burner, as there is
non-negligible radial in-flow of reactants towards the
center of the combustor, as was confirmed by LES calcu-
lations (Biagioli 2006). However, considering the whole
CZ, as it was defined above, we may assume that this in-
flow of reactants into CZ in transverse direction is small
compared to the in-flow in axial direction.
The experimentally determined flame position of some
section of the flame volume is defined as the average
z-coordinate zfl of the section considered, i.e. the CZ in our
case. All time-dependent quantities are assumed to be
phase-averaged with respect to the dominant acoustic
oscillation with the angular frequency x. The laboratory
flame propagation velocity _zflðtÞ is the difference between
the gas flow velocity vtotgas tð Þ at zfl(t), and the propagation
velocity vr(t), in opposite direction, of the flame relative to
the gas flow:
_zflðtÞ ¼ vtotgas zfl; tð Þ � vrðtÞ ðA1Þ
We assume furthermore that vr(t) oscillates sinusoidally
with the frequency x, the amplitude of the oscillation being
d�vr :
vrðtÞ ¼ �vr 1þ d cos xt � að Þ½ �: ðA2Þ
For thick flames—as in our case—vr(t) differs in general
from the burning velocity vb, which is defined as volume of
reactant gas burnt per unit of cross section and time unit.
The temporal averages of vr tð Þ � �vrð Þ and vb, however, are
equal.
180 Exp Fluids (2008) 45:167–182
123
The gas flow velocity vtotgas can be decomposed in a sta-
tionary part vgas,0, and the phase-dependent gas flow
velocity vvar(t):
vtotgas tð Þ ¼ vgas;0 þ vvar tð Þ: ðA3Þ
The stationary gas flow velocity vgas,0 depends on the axial
coordinate z. In our model case of a homogeneous flow
field, parallel to the axis, the flame is situated close, in
upstream direction, to the stagnation point of the gas flow.1
We approximate vgas,0(z) linearly:
vgas;0 zð Þ ¼ vgas;0 z0ð Þ þ z� z0ð Þv0gas;0 z0ð Þ� A0 � z� z0ð ÞA1: ðA4Þ
The parameters A0 and A1 are positive and defined by the
second equation above, and z0 is chosen so that the relation
A0 ¼ �vr holds. From the equations above, one obtains the
following equation:
_zfl tð Þ þ A1 zfl tð Þ � z0b c ¼ �vvar tð Þ � d�vr cos xt � að Þ: ðA5Þ
Equation (A5) is a differential equation for zfl, if the
parameters d�vr and a of the relative flame propagation
velocity and vvar(t) are known. On the other hand, if zfl and
vvar(t) are known experimentally, (A5) can be used to
determine the parameters d�vr and a.
The solution of Eq. (A5) with respect to zfl is straight-
forward in the case that the phase-dependent gas flow
velocity can be neglected. One obtains:
zfl ¼d�vr
x 1þ f2� �1=2
sin xt � uð Þ þ z0; ðA6Þ
with
f ¼ A1
x: ðA7Þ
The angle u is determined by
f sin xt � uð Þ þ cos xt � uð Þ¼ � 1þ f2
� �1=2cos xt � að Þ: ðA8Þ
Equation (A6) states that in an inhomogeneous flow field
the amplitude of the flame motion is shrunk by the factor
1þ f2� �1=2
; compared to the case of homogeneous flow.
We may distinguish two limiting cases:
1. f � 1. We obtain a & u + p. This means that the
maximum of the flame velocity relative to the gas is at
the phase of the backward zero crossing of the flame.
2. f � 1. We obtain a & u + (3p/2). This means that
the maximum of the flame velocity relative to the gas
is at the phase of the minimum flame position.
An approximation to Eq. (A6), valid for large f, is
obtained with the following simple argument: In general, at
the turning points of the flame motion, zfl,max and zfl,min, the
flame propagation velocity relative to the gas is equal to the
gas flow velocity. For large f, i.e. for large A1 compared to
x, we may assume that approximately the following
equation is true:
vgas;0 zfl;min
� �� vgas;0 zfl;max
� �¼ 2d�vr: ðA9Þ
From the definition of the derivative A1 in Eq. (A6)
follows
A1Dzfl ¼ 2d�vr; ðA10Þ
where by definition
Dzfl ¼ zfl;max � zfl;min: ðA11Þ
Thus, for f � 1 results, in agreement with Eq. (A6)
Dzfl ¼2d�vr
xf: ðA12Þ
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